Scilab Function
Last update : 13/12/2005

datafit - Parameter identification based on measured data

### Calling Sequence

  `[p,err]=datafit([imp,] G [,DG],Z [,W],[contr],p0,[algo],[df0,[mem]],`
  `[work],[stop],['in'])`

### Parameters

- `imp`: scalar argument used to set the trace mode. `imp=0` nothing (execpt errors) is reported, `imp=1` initial and final reports, `imp=2` adds a report per iteration, `imp>2` add reports on linear search. Warning, most of these reports are written on the Scilab standard output.
- `G`: function descriptor (e=G(p,z), e: ne x 1, p: np x 1, z: nz x 1)
- `DG`: partial of G wrt p function descriptor (optional; S=DG(p,z), S: ne x np)
- `Z`: matrix [z_1,z_2,...z_n] where z_i (nz x 1) is the ith measurement
- `W`: weighting matrix of size ne x ne (optional; defaut no ponderation)
- `contr`: `'b',binf,bsup` with `binf` and `bsup` real vectors with same dimension as `p0`. `binf` and `bsup` are lower and upper bounds on `p`.
- `p0`: initial guess (size np x 1)
- `algo`: `'qn'` or `'gc'` or `'nd'` . This string stands for quasi-Newton (default), conjugate gradient or non-differentiable respectively. Note that `'nd'` does not accept bounds on `x` ).
- `df0`: real scalar. Guessed decreasing of `f` at first iteration. (`df0=1` is the default value).
- `mem :` integer, number of variables used to approximate the Hessian, (`algo='gc' or 'nd'`). Default value is around 6.
- `stop`: sequence of optional parameters controlling the convergence of the algorithm. `stop= 'ar',nap, [iter [,epsg [,epsf [,epsx]]]]`
  - `"ar"` : reserved keyword for stopping rule selection defined as follows:
  - `nap`: maximum number of calls to `fun` allowed.
  - `iter`: maximum number of iterations allowed.
  - `epsg`: threshold on gradient norm.
  - `epsf`: threshold controlling decreasing of `f`
  - `epsx`: threshold controlling variation of `x`. This vector (possibly matrix) of same size as `x0` can be used to scale `x`.
- `"in"` : reserved keyword for initialization of parameters used when `fun` in given as a Fortran routine (see below).
- `p`: Column vector, optimal solution found
- `err`: scalar, least square error.

### Description

`datafit` is used for fitting data to a model. For a given function `G(p,z)`, this function finds the best vector of parameters `p` for approximating `G(p,z_i)=0` for a set of measurement vectors `z_i`. Vector `p` is found by minimizing `G(p,z_1)'WG(p,z_1)+G(p,z_2)'WG(p,z_2)+...+G(p,z_n)'WG(p,z_n)`

`datafit` is an improved version of `fit_dat`.

### Examples

    //generate the data
    function y=FF(x,p),y=p(1)*(x-p(2))+p(3)*x.*x,endfunction
    X=[];Y=[];
    pg=[34;12;14] //parameter used to generate data
    for x=0:.1:3, Y=[Y,FF(x,pg)+100*(rand()-.5)];X=[X,x];end
    Z=[Y;X];

    //The criterion function
    function e=G(p,z),
       y=z(1),x=z(2);
       e=y-FF(x,p),
    endfunction

    //Solve the problem
    p0=[3;5;10]
    [p,err]=datafit(G,Z,p0);

    scf(0);clf()
    plot2d(X,FF(X,pg),5) //the curve without noise
    plot2d(X,Y,-1)  // the noisy data
    plot2d(X,FF(X,p),12) //the solution

    //the gradient of the criterion function
    function s=DG(p,z),
       a=p(1),b=p(2),c=p(3),y=z(1),x=z(2),
       s=-[x-b,-a,x*x]
    endfunction

    [p,err]=datafit(G,DG,Z,p0);
    scf(1);clf()
    plot2d(X,FF(X,pg),5) //the curve without noise
    plot2d(X,Y,-1)  // the noisy data
    plot2d(X,FF(X,p),12) //the solution

### See Also

`lsqrsolve`, `optim`, `leastsq`,